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1984 | Buch

Riesz’s Lemma and Compactness in Banach Spaces Kapitel lesen Erstes Kapitel lesen

Sequences and Series in Banach Spaces

verfasst von: Joseph Diestel

Verlag: Springer New York

Buchreihe : Graduate Texts in Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

This volume presents answers to some natural questions of a general analytic character that arise in the theory of Banach spaces. I believe that altogether too many of the results presented herein are unknown to the active abstract analysts, and this is not as it should be. Banach space theory has much to offer the prac­ titioners of analysis; unfortunately, some of the general principles that motivate the theory and make accessible many of its stunning achievements are couched in the technical jargon of the area, thereby making it unapproachable to one unwilling to spend considerable time and effort in deciphering the jargon. With this in mind, I have concentrated on presenting what I believe are basic phenomena in Banach spaces that any analyst can appreciate, enjoy, and perhaps even use. The topics covered have at least one serious omission: the beautiful and powerful theory of type and cotype. To be quite frank, I could not say what I wanted to say about this subject without increasing the length of the text by at least 75 percent. Even then, the words would not have done as much good as the advice to seek out the rich Seminaire Maurey-Schwartz lecture notes, wherein the theory's development can be traced from its conception. Again, the treasured volumes of Lindenstrauss and Tzafriri also present much of the theory of type and cotype and are must reading for those really interested in Banach space theory.

Inhaltsverzeichnis

Frontmatter
Chapter I. Riesz’s Lemma and Compactness in Banach Spaces
Abstract
In this chapter we deal with compactness in general normed linear spaces. The aim is to convey the notion that in normed linear spaces, norm-compact sets are small—both algebraically and topologically.
Joseph Diestel
Chapter II. The Weak and Weak* Topologies: An Introduction
Abstract
As we saw in our brief study of compactness in normed linear spaces, the norm topology is too strong to allow any widely applicable subsequential extraction principles. Indeed, in order that each bounded sequence in X have a norm convergent subsequence, it is necessary and sufficient that X be finite dimensional. This fact leads us to consider other, weaker topologies on normed linear spaces which are related to the linear structure of the spaces and to search for subsequential extraction principles therein. As so often happens in such ventures, the roles of these topologies are not restricted to the situations initially responsible for their introduction. Rather, they play center court in many aspects of Banach space theory.
Joseph Diestel
Chapter III. The Eberlein-Šmulian Theorem
Abstract
We saw in the previous chapter that regardless of the normed linear space X, weak* closed, bounded sets in X* are weak* compact. How does a subset K of a Banach space X get to be weakly compact? The two are related. Before investigating their relationship, we look at a couple of necessary ingredients for weak compactness and take a close look at two illustrative nonweakly compact sets.
Joseph Diestel
Chapter IV. The Orlicz-Pettis Theorem
Abstract
In this chapter we prove the following theorem.
Joseph Diestel
Chapter V. Basic Sequences
Abstract
In any earnest treatment of sequences and series in Banach spaces a featured role must be reserved for basic sequences. Our initial discussion of this important notion will occupy this whole chapter. A foundation will be laid on which we will build several of the more interesting constructs in the theory of sequences and series in Banach spaces.
Joseph Diestel
Chapter VI. The Dvoretsky-Rogers Theorem
Abstract
Recall that a normed linear space X is a Banach space if and only if given any absolutely summable series in ∑n x n in X, lim n ∑ k-1 n x k exists. Of course, in case X is a Banach space, this gives the following implication for a series ∑ n x n : if ∑ n ∥x n ∥ < ∞, then ∑ n x n is unconditionally convergent; that is, ∑ n x π(n) converges for each permutation π of the natural numbers.
Joseph Diestel
Chapter VII. The Classical Banach Spaces
Abstract
To this juncture, we have dealt with general theorems concerning the nature of sequential convergence and convergence of series in Banach spaces. Many of the results treated thus far were first derived in special cases, then understood to hold more generally. Not too surprisingly, along the path to general results many important theorems, special in their domain of applicability, were encountered. In this chapter, we present more than a few such results.
Joseph Diestel
Chapter VIII. Weak Convergence and Unconditionally Convergent Series in Uniformly Convex Spaces
Abstract
In this chapter, we prove three results too stunning not to be in the spotlight. These results are typical of the most attractive aspects of the theory of Banach spaces in that they are proved under easily stated, commonly understood hypotheses, are readily appreciated by Banach spacers and non-Banach spacers alike, and have proofs that bare their geometric souls.
Joseph Diestel
Chapter IX. Extremal Tests for Weak Convergence of Sequences and Series
Abstract
This chapter has two theorems as foci. The first, due to the enigmatic Rainwater, states that for a bounded sequence (x n) in a Banach space X to converge weakly to the point x, it is necessary and sufficient that x*x = lim n x*x n hold for each extreme point x* of B x* . The second improves the Bessaga-Pelczynski criterion for detecting c 0’s absence; thanks to Elton, we are able to prove that in a Banach space X without a copy of c 0 inside it, any series ∑ n x n for which ∑n∣x*x n∣ < ∞ for each extreme point x* of B x* is unconditionally convergent.
Joseph Diestel
Chapter 10. Grothendieck’s Inequality and the Grothendieck-Lindenstrauss-Pelczynski Cycle of Ideas
Abstract
In this section we prove a profound inequality due, as the section title indicates, to Grothendieck. This inequality has played a fundamental role in the recent progress in the study of Banach spaces. It was discovered in the 1950s, but its full power was not generally realized until the late 1960s when Lindenstrauss and Pelczynski, in their seminal paper “Absolutely summing operators in ℒp spaces and their applications,” brutally reminded functional analysts of the existence and importance of the powerful ideas and work of Grothendieck. Since the Lindenstrauss-Pelczynski paper, the Grothendieck inequality has seen many proofs; in this, it shares a common feature of most deep and beautiful results in mathematics. The proof we present is an elaboration of one presented by R. Rietz. It is very elementary.
Joseph Diestel
An Intermission: Ramsey’s Theorem
Abstract
Some notation, special to the present discussion, ought to be introduced. If A and B are subsets of the set N of natural numbers, then we write A < B whenever a < b holds for each a ∈ A and b ∈ B. The collection of finite subsets of A is denoted by ℘ <∞(A) and the collection of infinite subsets of A by ℘ ∞(A). More generally for A,B ⊆ N we denote by ℘ <∞(A,B the colelction { X ∈ ℘ <∞ (N) : A ⊆ X A ∪ B, A < X\ A } and by ℘ ∞ (A,B) the collection { X ∈ ℘ ∞ (N): A ⊆ X ⊆ A ∪ B, A < X\ A }.
Joseph Diestel
Chapter XI. Rosenthal’s l 1 Theorem
Abstract
The Eberlein-Šmulian theorem tells us that in order to be able to extract from each bounded sequence in X a weakly convergent subsequence it is both necessary and sufficient that X be reflexive. Suppose we ask less. Suppose we ask only that each bounded sequence in X have a weakly Cauchy subsequence. [Recall that a sequence (x n ) in a Banach space X is weakly Cauchy if for each x* ∈ X* the scalar sequence (x*x n ) is convergent.] When can one extract from each bounded sequence in X a weakly Cauchy subsequence?
Joseph Diestel
Chapter XII. The Josefson-Nissenzweig Theorem
Abstract
From Alaoglu’s theorem and the F. Riesz theorem, we can conclude that for infinite-dimensional Banach spaces X the weak* topology and the norm topology in X* differ. Can they have the same convergent sequences? The answer is a resounding “no!” and it is the object of the present discussion. More precisely we will prove the following theorem independently discovered by B. Josef son and A. Nissenzweig.
Joseph Diestel
Chapter XIII. Banach Spaces with Weak Sequentially Compact Dual Balls
Abstract
Alaoglu’s theorem ensures that every bounded sequence (x n * ) in X* has a weak* convergent subnet. When can one actually extract a weak* convergent subsequence? As yet, no one knows. In this chapter a few of the most attractive conditions assuring the existence of such subsequences are discussed.
Joseph Diestel
Chapter XIV. The Elton-Odell (1 + ε)-Separation Theorem
Abstract
In this concluding chapter we prove the following separation theorem, due to J. Elton and E. Odell.
Joseph Diestel
Backmatter
Metadaten
Titel
Sequences and Series in Banach Spaces
verfasst von
Joseph Diestel
Copyright-Jahr
1984
Verlag
Springer New York
Electronic ISBN
978-1-4612-5200-9
Print ISBN
978-1-4612-9734-5
DOI
https://doi.org/10.1007/978-1-4612-5200-9